Structure of Lower Tails in Sparse Random Graphs
Name
Random Struct Algorithms - 2025 - Chin - Structure of Lower Tails in Sparse Random Graphs.pdf
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261.07 KB
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Adobe PDF
Checksum (MD5)
7d559b036418bc57e7c122a820747fda
Author(s)
Chin, Byron
Date Issued
August 11, 2025
Journal
Random Structures & Algorithms
Publisher
Wiley
Citation
Chin, B. 2025. “ Structure of Lower Tails in Sparse Random Graphs.” Random Structures & Algorithms 67, no. 1: e70028.
Version
Final published version
Abstract
We study the typical structure of a sparse Erdős–Rényi random graph conditioned on the lower tail subgraph count event. We show that in certain regimes, a typical graph sampled from the conditional distribution resembles the entropy minimizer of the mean field approximation in the sense of both subgraph counts and cut norm. The main ingredients are an adaptation of an entropy increment scheme of Kozma and Samotij, and a new stability for the solution of the associated entropy variational problem. The proof can be interpreted as a structural application of the new probabilistic hypergraph container lemma for sparser than average sets, and suggests a more general framework for establishing such typical behavior statements.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
Creative Commons Attribution
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DOI of Published Version
https://doi.org/10.1002/rsa.70028
